A Numerical Renormalization Group for Low Dimensional Field Theories

本程序将扩展和应用PI和CO-PI发展的数值技术，利用该技术可以以非微扰的方式研究广泛的低维相互作用场论的性质。该方法以一类强相互作用零维和一维系统的可积场和共形场理论处理产生的谱和矩阵元素的形式，将已有的数据作为输入。以此为基础，通过重整化群过程引导的Hilbert空间的受控截断，对通过扰动或耦合建立的更复杂的场论进行了数值分析。这就是重整化群改进截断谱方法(RGTSA)。最强的相互作用在一开始就被考虑在内。利用它可以研究1+1维可积场论和共形场论的任意微扰，以及研究一维耦合场论的大阵列，从而允许将该方法扩展到2+1维。数值重整化群技术借鉴了量子杂质问题的研究和密度矩阵重整化群方法来研究一维晶格模型。这项技术具有足够的通用性，可以提取光谱、关联函数和非平衡现象，如量子猝灭后的动力学。这项技术能够表征真实材料的性质。作为对该技术原理的验证，对半导体碳纳米管的激子谱进行了研究，结果表明该模型能很好地描述实验数据。这项技术将被用于研究混合纳米系统的光学性质，目的是更好地了解这种混合系统作为太阳能设备的潜力。两名研究生将接受低维量子场理论及其在凝聚态环境中的应用方面的广泛培训。他们将进行原创性研究，开发具有跨领域适用性的理论方法。研究结果将发表在期刊上，并通过研讨会、座谈会和会议报告等形式公布。

This program will extend and apply a numerical technique developed by the PI and CO-PI by which the properties of a wide range of interacting field theories in low dimensions can be studied in a nonperturbative fashion. The approach takes as input already available data, in the form of spectra together with matrix elements, arising from integrable and conformal field theoretical treatments of a wide class of strongly interacting zero and one dimensional systems. Using this as a foundation, more complicated field theories, built by either perturbing or coupling together such systems, are analyzed numerically through a controlled truncation of the Hilbert space guided by a renormalization group procedure. This is the renormalization group improved truncated spectrum approach (RGTSA). The strongest interactions are taken into account at the very start. Using it one can study arbitrary perturbations of 1+1 dimensional integrable and conformal field theories as well as study large arrays of coupled one-dimensional field theories, permitting the extension of the approach to 2+1 dimensions. The numerical renormalization group technique borrows ideas from the study of quantum impurity problems and density matrix renormalization group approaches to one dimensional lattice models. This technique is sufficiently versatile to extract the spectrum, correlation functions and non-equilibrium phenomena such as the dynamics following a quantum quench. This technique is able to characterize the properties of real materials. As a proof of principle of the technique, the excitonic spectrum of semi-conducting carbon nanotubes has been studied and it has been shown that the modeling well describes experimental data. This technique will be used to study the optical properties of hybrid nanosystems with the aim of better understanding this hybrid's potential as a solar energy device. Two graduate students will receive extensive training in low dimensional quantum field theories as well as their application in condensed matter settings. They will conduct original research and develop theoretical methods with cross-field applicability. The results will be presented in journals and through seminars, colloquia, and conference presentations.